@article {IOPORT.06039150,
    author       = {Conidis, Chris J.},
    title        = {Effectively approximating measurable sets by open sets.},
    year         = {2012},
    journal      = {Theoretical Computer Science},
    volume       = {428},
    issn         = {0304-3975},
    pages        = {36-46},
    publisher    = {Elsevier Science Publishers, Amsterdam},
    doi          = {10.1016/j.tcs.2012.01.011},
    abstract     = {Summary: We examine an effective version of the standard fact from analysis which says that, for any $\varepsilon >0$ and any Lebesgue-measurable subset of Cantor space, $X\subseteq 2^{\omega}$, there is an open set $U_{\varepsilon} \subseteq 2^{\omega}$, $U_{\varepsilon} \supseteq X$, such that $\mu (U_{\varepsilon})\leq \mu (X)+\varepsilon $, where $\mu (Z)$ denotes the Lebesgue measure of $Z \subseteq 2^{\omega }$, that arises naturally in the context of algorithmic randomness. More specifically, our main result shows that for any given rational numbers $0 \leq \varepsilon <\varepsilon ^{\prime}\leq 1$, and uniformly computably enumerable sequence $\{U_{n}\}_{n \in \omega}$ of $\Sigma_1^0$-classes such that ($\forall n)[\mu (U_{n})\leq \varepsilon]$, there exists a $\Sigma_1^{0, \emptyset^{\prime}}$-class, $Y$, such that $Y \supseteq \lim \inf_{n}U_{n}$, and $\mu (Y)\leq \varepsilon^{\prime}$. Moreover, $Y$ can be obtained uniformly from $\varepsilon$, $\varepsilon^{\prime}$, and a u.c.e. index for $\{U_{n}\}_{n \in \omega}$. This answers a recent question of Bienvenu, Muchnik, Shen, and Vereshchagin. We also determine the truth-values of several modifications of our main result, showing that several similar, but stronger, statements are false.},
    identifier   = {06039150},
}